Define $\displaystyle b_n$ as $\displaystyle b_0 = 0$ , $\displaystyle b_1 = 3$ and $\displaystyle b_n = 7b_{n-1}-10b_{n-2}$. Prove by induction that $\displaystyle b_n=5^n-2^n$

Proof:

Let P(n) be the propositional statement for n>1 "$\displaystyle 5^n-2^n=7b_{n-1}-10b_{n-2}$"

The LHS of P(2) is 5^2-2^2=21 and the RHS if P(2) is 7(3)-10(0)=21. So P(2) is true.

Now assume that P(j) is true for all j<k. That is assume $\displaystyle 5^j-2^j=7b_{j-1}-10b_{j-2}$. Then by this assumption that P(j-1) can be written as $\displaystyle 7(5^{k-1}-2^{k-1})-10(5^{k-2}-2^{k-2})$

Now I'm kind of stuck. This is either going to take some sick algebra or I made a boo-boo.